Length complexity of tensor products

TL;DR

This paper develops techniques to estimate the torsion in tensor products of finitely generated modules over Noetherian rings, focusing on bounds related to global invariants for specific module classes like vector bundles and low-dimensional modules.

Contribution

It introduces new methods to bound torsion in tensor products based on global invariants, diverging from previous rigidity-focused studies.

Findings

Established bounds for torsion in tensor products of vector bundles.

Derived torsion estimates for modules of dimension at most three.

Provided a framework for analyzing tensor product torsion in specific module classes.

Abstract

In this paper we introduce techniques to gauge the torsion of the tensor product $A\otimes_RB$ of two finitely generated modules over a Noetherian ring $R$. The outlook is very different from the study of the rigidity of Tor carried out in the work of Auslander and other authors. Here the emphasis in on the search for bounds for the torsion part of $A\otimes_R B$ in terms of global invariants of $A$ and of $B$ in special classes of modules: vector bundles and modules of dimension at most three.